Part 4: Data Analysis

Question 1

Define central tendency.

Answer 1

Central tendency are the values in the center of data along a number line, which include the mean, the median, and the mode.

Question 2

Define mean.

Answer 2

The arithmetic average.

Question 3

Define median.

Answer 3

The value of the piece of data in the exact middle of a data series.

Question 4

Define mode.

Answer 4

The value that occurs the most frequently in a list.

Question 5

What is a measure of position?

Answer 5

It is the categorization of data in ordered groups, like percentiles and quartiles.

Question 6

How do you determine a median in set containing an even number of values?

Answer 6

Average the two middle values.

Question 7

What is a measure of dispersion?

Answer 7

It is the degree of spread in the data, including range, the interquartile range, and the standard deviation.

Question 8

What is the range?

Answer 8

The difference between to two outliers in a set.

Question 9

What is the interquartile range?

Answer 9

The difference between Q1 and Q3.

Question 10

What is standard deviation?

Answer 10

It is the amount that each data differs from the mean. So, the greater the range, the greater the standard deviation.

Question 11

How is standard deviation calculated?

Answer 11

1. Find the mean.
2. Find the difference between the mean and each value.
3. square each difference.
4. find the average of each squared difference.
5. Take the nonnegative square root of the average of the squared differences.

Question 12

What is standardization?

Answer 12

It is the process where you subtract the mean then divide by the standard deviation to determine how many standard deviations away a piece of data is from the mean.

Question 13

What are some characteristics of sets?

Answer 13

Non repeated and non ordered values.

Question 14

If a set is represented by S, how do you represent the number of values in a set?

Answer 14

|S|

Question 15

How is an empty set represented.

Answer 15

ø

Question 16

What is the multiplication principle?

Answer 16

If there are k choices for the 1st object and m choice for the 2nd object, then there are km choices for their combination.

Question 17

What is a permutation?

Answer 17

It is the number of ways n objects can be ordered, determined by multiplying n(n-1)(n-2)…1. Also representing as n!

Question 18

How do you determine a permutation for a subset of objects? Where n is the number of objects in the set, and k is the number of items.

Answer 18

n!/(n - k!)

Question 19

What is a combination?

Answer 19

It is selection of a set without worrying about ordering the items.

Question 20

What is the formula for determining a combination?

Answer 20

n!/k!(n-k)!

Question 21

What are the two different notations for selecting k objects from n?

Answer 21

nCk and (n k)

Question 22

What is probability rule #1 for events E and F? (or)

Answer 22

P(E or F) = P(E) + P(F) - P (E and F)

Question 23

What is probability rule #2 for events E and F? (mutually exclusive or)

Answer 23

If E and F are mutually exclusive, then the P(EorF) becomes P(E) + P(F)

Question 24

Probability rule #3 for events E and F? (and)

Answer 24

If E and F are independent, then P(EandF) = P(E)P(F)

Question 25

Exercise 1. The daily temperatures, in degrees Fahrenheit, for 10 days in May were 61, 62, 65, 65, 65, 68, 74, 74, 75, and 77.

(a) Find the mean, median, mode, and range of the temperatures.
(b) If each day had been 7 degrees warmer, what would have been the mean, median, mode, and range of those 10 temperatures?

Answer 25

Mean: 68.6 degrees
Median: 66.5 degrees
Mode: 65 degrees
Range: 16

b) 75.6, 73.5, 72, 16

Question 26

Exercise 2. The numbers of passengers on 9 airline flights were 22, 33, 21, 28, 22, 31, 44, 50, and 19. The standard deviation of these 9 numbers is approximately equal to 10.22.

(a) Find the mean, median, mode, range, and interquartile range of the 9 numbers.
(b) If each flight had had 3 times as many passengers, what would have been the mean, median, mode, range, interquartile range, and standard deviation of the 9 numbers?
(c) If each flight had had 2 fewer passengers, what would have been the interquartile range and standard deviation of the 9 numbers?

Answer 26

19, 21, 22, 22, 28, 31, 33, 44, 50
Mean: 30
Median: 28
Mode: 22
Range: 31
Interquartile Range: 38.5 - 21.5 = 17
Difference from Mean:
8, 3, 9, 2, 8, 1, 14, 20, 11
Squares:
64, 9, 81, 4, 64, 1, 196, 400, 121
Average of the squares: 940/9 = 104.44444
Square root of the average:
10.22
Standard Deviation is 10.22

3x: 57, 63, 66, 66, 84, 93, 99, 132, 150
Mean: 90
Median: 84
Mode: 66
Range: 93
Interquartile Range: 115.5 - 64.5 = 51
Difference from the mean:
-24, 9, -27, -6, -24, 3, 42, 60, -33
Square of the differences:
576, 81, 729, 36, 576, 9, 1764, 3600, 1089
Average of the squares: 940
Square root of the average: 30.66
Standard Deviation: 30.66

Two Fewer: 20, 31, 19, 26, 20, 29, 42, 48, 17
Interquartile Range: 17
Standard Deviation: 10.22

Question 27

Exercise 4. Find the mean and median of the values of the random variable X, whose relative frequency distribution is given in the following table.

0: 0.18
1: 0.33
2: 0.10
3: 0.06
4: 0.33

Answer 27

Mean: 2.03
Median: 1

Question 28

Eight hundred insects were weighed, and the resulting measurements, in milligrams, are summarized in the following boxplot.
Data Analysis Figure 20
(a) What are the range, the three quartiles, and the interquartile range of the measurements?
(b) If the 80th percentile of the measurements is 130 milligrams, about how many measurements are between 126 milligrams and 130 milligrams?

Answer 28

Range: 146 - 105 = 41
Q1: 114
Q2: 118
Q3: 126
Interquartile Range: 126-114 = 12
b) 800 * 0.05 = 40

Question 29

In how many different ways can the letters in the word STUDY be ordered?

Answer 29

5! = 5 * 4 * 3 * 2 * 1 = 120

Question 30

Exercise 7. Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row of 5 seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?

Answer 30

4! = 4 * 3 * 2 * 1 = 24

Question 31

Exercise 8. How many 3-digit positive integers are odd and do not contain the digit 5 ?

Answer 31

8 * 9 * 4 = 288

Question 32

Exercise 9. From a box of 10 lightbulbs, you are to remove 4. How many different sets of 4 lightbulbs could you remove?

Answer 32

1 (gah!)

10!/4!(10 - 4!) = 10 * 9 * 8 * 7 / 24 = 210

Question 33

Exercise 10. A talent contest has 8 contestants. Judges must award prizes for first, second, and third places, with no ties.

(a) In how many different ways can the judges award the 3 prizes?
(b) How many different groups of 3 people can get prizes?

Answer 33

8 * 7 * 6 = 336

336/6 = 56

Question 34

Exercise 11. If an integer is randomly selected from all positive 2-digit integers, what is the probability that the integer chosen has

(a) a 4 in the tens place?
(b) at least one 4 in the tens place or the units place? (c) no 4 in either place?

Answer 34

a) 1/9
b) P(AorB) = 10/90 + 9/90 = 18/90 = 1/5
c) 4/5

Question 35

Exercise 12. In a box of 10 electrical parts, 2 are defective.
(a) If you choose one part at random from the box, what is the probability that it is not
defective?
(b) If you choose two parts at random from the box, without replacement, what is the probability that both are defective?

Answer 35

a) 4/5

b) 1/5 * 1/9 = 1/45

Question 36

Exercise 13. A certain college has 8,978 full-time students, some of whom live on campus and some of whom live off campus.
The following table shows the distribution of the 8,978 full-time students, by class and living arrangement.

Freshmen
Sophomores

Juniors

Seniors

Live on campus
1,812
1,236
950
542
Live off campus

625

908
1,282

1,623

(a) If one full-time student is selected at random, what is the probability that the student who is chosen will not be a freshman?
(b) If one full-time student who lives off campus is selected at random, what is the probability that the student will be a senior?
(c) If one full-time student who is a freshman or sophomore is selected at random, what is the probability that the student will be a student who lives on campus?

Answer 36

a) 8978 -1812 - 625)/ 8978 = 73%
b) 1623/ 1623 + 1282 + 908 + 625 = 36.6%
c) 1812 + 1236 / 1812 + 1236 + 625 + 908 = 66.5%

Question 37

Exercise 14. Let A, B, C, and D be events for which P(AorB)=0.6, P(A)=0.2, P(CorD)=0.6, and P(C)=0.5.
The events A and B are mutually exclusive, and the events C and D are independent.
(a) Find P(B).
(b) Find P(D).

Answer 37

P(B) = 0.4
0.2 = P(D)

Question 38

Exercise 15. Lin and Mark each attempt independently to decode a message. If the probability that Lin will decode the message is 0.80 and the probability that Mark will decode the message is 0.70, find the probability that

(a) both will decode the message
(b) at least one of them will decode the message
(c) neither of them will decode the message

Answer 38

a) .8 * .7 = 0.56
b) .8 + .7 - .56 = 1.50 - .56 = 0.94
c) 0.2 * 0.3 = 0.06

Question 39

Exercise 16. Data Analysis Figure 21 below shows the graph of a normal distribution with mean m and standard deviation d, including approximate percents of the distribution corresponding to the six regions shown.
Data Analysis Figure 21
Suppose the heights of a population of 3,000 adult penguins are approximately normally distributed with a mean of 65 centimeters and a standard deviation of 5 centimeters.
(a) Approximately how many of the adult penguins are between 65 centimeters and 75 centimeters tall?
(b) If an adult penguin is chosen at random from the population, approximately what is the probability that the penguin’s height will be less than 60 centimeters? Give your answer to the nearest 0.05.

Answer 39

a) 3000 * 0.48 = 1440

b) 16%

Question 40

Exercise 17. This exercise is based on the following graph.
Data Analysis Figure 22
(a) For which year did total expenditures increase the most from the year before?
(b) For 2001, private school expenditures were what percent of total expenditures? Give your answer to the nearest percent.

Answer 40

a) 1998

b) 30/160 = 19%

Question 41

Exercise 18. This exercise is based on the following data.
Data Analysis Figure 23
(a) In 2001, how many categories each comprised more than 25 million workers?
(b) What is the ratio of the number of workers in the Agricultural category in 2001 to the projected number of such workers in 2025 ?
(c) From 2001 to 2025, there is a projected increase in the number of workers in which of the following three categories?
Category1: Sales
Category2: Service
Category3: Clerical

Answer 41

a) agriculture, manufacturing, clerical
b) 150mil * .18 : 175 mil * .24 = 27 mil : 42 mil = 9: 14
c) Service , Sales, Clerical (All)

Question 42

Exercise 19. This exercise is based on the following data.
Data Analysis Figure 24
(a) In 2003 the family used a total of 49 percent of its gross annual income for two of the categories listed. What was the total amount of the family’s income used for those same categories in 2004 ?
(b) Of the seven categories listed, which category of expenditure had the greatest percent increase from 2003 to 2004 ?

Answer 42

a) real estate and savings

b) Miscellaneous